Monday, July 19, 2010

Fracture Mechanics in Turbine Blade Analysis

I would like to take some time to start a discussion on fracture mechanics. The calculations for a basic fracture mechanics analysis are fairly simple, but they can play a very important role in failure analyses. This topic comes up quite frequently in turbine blade work.

So what is fracture mechanics? In short it’s a method of determining the time it takes a crack in a part to grow to failure under a specific loading condition. The crack growth stage of fatigue can make up a significant portion of a products life. This happens in products ranging from bicycles, to airplanes, to steam turbine blades.

At the heart of fracture mechanics is the stress intensity factor K defined as:

Where:

f(g) is a correction factor based on crack geometry. This value tends to be between 1 and 1.4.

a is the crack length

s is the remote stress

Fatigue crack growth is divided into 3 regions as shown in the figure below. In this figure, crack growth rate (da/dN) is plotted on the vertical axis in log scale and Stress Intensity Range (DK=Kmax-Kmin) is plotted on the horizontal axis in log scale. Region I is associated with crack threshold effects (the area where a crack first begins to grow), Region II is an area of linear growth (Paris region), and Region III exhibts extremely high/unstable crack growth.

For design purposes the focus is on Regions I and II. Crack growth is so fast in Region III that it does not have a significant effect on the total crack propagation life. Noted on the graph is DKth, the threshold stress intensity which is determined through testing. This value marks the beginning of crack growth. Kc is the critical stress intensity and values higher than this predict fracture.

When performing a turbine blade analysis we often want to determine if a dynamic stress condition is severe enough to grow a crack (Refer to previous posts on blade analysis). To determine that, we run a fracture mechanics analysis for Region I of the above graph. If the stress condition and initial flaw size is not capable of growing a crack then we need not be concerned with removing the near resonant condition.

This analysis starts with calculating the stress intensity factor range:

In the case of an edge crack on a turbine blade airfoil the a correction factor f(g)=1.12 is typically used. Ds is the stress range, or dynamic stress for turbine blades (again refer to my last post on dynamic stress analysis). If DK is greater than DKth, then a crack will propagate under the given loading condition.

One other thing to consider is the R ratio. Test results for DKth values are very dependant on the conditions which they were tested at. A particular DKth value will only apply to a loading condition that has the same R ratio as the test. The R ratio calculation is shown below:

Where sm is the mean steady stress and sd is the alternating or dynamic stress. If the R ratio for the DKth test is different than calculated above, the DKth will need to be adjusted to account for the difference. One common method for the compensation of DKth is Walker’s Equation:

Where is the value at R=0, g is a material constant and is typically between 0.3 and 1. Steels are typically around 0.5. Using this relationship, and assuming that is a constant, you can calculate the DKth value for any R ratio.

Thanks for reading,